Try Arduino or Picaxe web sites...heaps of tutorials to get you started and keep you going for many years.
What "specifically" is the project if you want better direction.
Most likely a capacitor but no markings anywhere so your guess is as good as anyone elses.
What makes you think it is faulty if that is the reason for the query.
Thought about fixing the old one....?
Other than that there are many keyfob remotes available but on the 400 or 900 mhz ...only a couple of dollars for complete setup....would need to convert the relay output to something usable though ....
What does the existing system consist of and how does...
If original was 6 volt and the new one is giving 15vac then clearly it is NOT the same.
From your last statement, it is clear you need the local assistance of someone with some basic meter reading and electronics skills.
Find it difficult to follow your logic...if zero out on the transformer and you have a new one, did you fit and test that......???
That aside, new one appears to have multiple outputs...is it the correct replacement...???
Nor will addition of additional devices at uC level to protect against electric shock....Assume you require to protect the user, then use off the shelf safety devices on the mains.